N=3 Supersymmetric Extension of KdV Equation

TL;DR

This paper constructs and analyzes N=3 supersymmetric extensions of the KdV equation, identifying a non-integrable family and proposing a modified version that may be integrable, with implications for superconformal algebra structures.

Contribution

It introduces a new modified N=3 super KdV equation with higher conserved quantities, suggesting potential integrability and connecting to superconformal algebra frameworks.

Findings

The original N=3 supersymmetric extension is non-integrable.

A modified N=3 super KdV equation with conserved quantities is proposed.

Reduction to N=2 yields a known 'would-be integrable' super KdV version.

Abstract

We construct a one-parameter family of N=3 supersymmetric extensions of the KdV equation as a Hamiltonian flow on N=3 superconformal algebra and argue that it is non-integrable for any choice of the parameter. Then we propose a modified N=3 super KdV equation which possesses the higher order conserved quantities and so is a candidate for an integrable system. Upon reduction to N=2, it yields the recently discussed ``would-be integrable'' version of the N=2 super KdV equation. In the bosonic core it contains a coupled system of the KdV type equation and a three-component generalization of the mKdV equation. We give a Hamiltonian formulation of the new N=3 super KdV equation as a flow on some contraction of the direct sum of two N=3 superconformal algebras.